For more information about this meeting, contact Becky Halpenny, Stephanie Zerby.

Title: | "The Generalized Finite Element Method: Numerical Treatment of Singularities, Interfaces, and Boundary Conditions" |
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Seminar: | Ph.D. Thesis Defense |

Speaker: | Qingqin Qu, Advisers: Anna Mazzucato/Victor Nistor, Penn State |

Abstract: | |

This dissertation is devoted to use the Generalized Finite Element Method to solve partial dierential equations numerically. As an extension of the standard Finite Element Method (FEM), the Generalized Finite Element Method (GFEM) is especially convenient for dealing with complicated do- mains, corner singularities, transmission problems and mixed boundary con- ditions. The GFEM is related to other methods, such as the hp cloud method and the extended nite element method. The GFEM diers from the stan- dard FEM in the construction of the nite-dimensional space in which the approximate solution is sought. Instead of using piecewise polynomials on each element of a triangulation of the domain, we dene the element spaces by using partitions of unity to combine the local approximation spaces, de- ned in each patch of the partition, in order to obtain the global GFEM space. In particular, the GFEM is an example of a meshless method, since the partition of unity need not be subordinated to a particular mesh. The GFEM allows one to include a priori knowledge about the local behavior of the solution, and gives the option of constructing trial spaces of any desired regularity. For transmission (interface) problems on domains with smooth, curved boundaries, we establish quasi-optimal rates of convergence of the numerical solution to the true solution by using a non-conforming Generalized Finite Element Method. A sequence of approximation spaces Sn are constructed that satisfy the following two conditions: (1) nearly zero boundary and inter- face matching, (2) approximability. The numerical solution is then obtained as a Galerkin approximation to the true solution. We prove that the ap- proximation error of order O(dim(Sn) m=2 ), where dim(Sn) is the dimension of the GFEM space Sn, and m is the degree of polynomials used for the local approximation of the solution. Numerical experiments are presented to demonstrate these theoretical results. For the case of singular domains, we study the GFEM approximation to solutions of the Poisson's problem in polygons. It is well-known that the loss of regularity of the exact solution due to domain singularities will deteriorate the convergence rate of the standard FEM if one uses quasi- uniform meshes. To circumvent the loss of regularity, we pose the problem in certain weighted Sobolev spaces, and show that the continuous problem has the expected regularity in these spaces. We construct GFEM approximation spaces using again a partition of unit and local approximation spaces. For the former, we use dilation techniques to deal with corner singularities, while we use standard piecewise polynomial spaces for the latter. We then establish quasi-optimal rate of convergence of the GFEM approximation to the exact solution both in weighted Sobolev spaces and then in Hilbert spaces. |

### Room Reservation Information

Room Number: | MB106 |
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Date: | 05 / 15 / 2012 |

Time: | 02:00pm - 04:00pm |